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Euler Sums Revisited

Apostol, Tom M. (2003) Euler Sums Revisited. In: Mathematical Properties of Sequences and Other Combinatorial Structures. Springer International Series in Engineering and Computer Science. No.726. Springer US , Boston, MA, pp. 121-132. ISBN 9781461350132. https://resolver.caltech.edu/CaltechAUTHORS:20190909-133031057

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Abstract

A large literature exists relating Riemann’s zeta function ζ(s)=∑k=1 ∞ 1/ks,R(s)>1, and partial sums of the harmonic series, h(n)=∑k=1 n 1/k. Much of the research originated from two striking formulas discovered by Euler, ∑n=1 ∞ h(n)/n^2=2ζ(3) (1) , ∑n=1 ∞ h(n)/n^3=54ζ(4), (2) and a recursion formula, also due to Euler, which states that for integer a ≥2 we have (a+1/2)ζ(2a)=∑k=1 a−1 ζ(2k)ζ(2a−2k). (3) For a = 2 and a =3 this gives ζ(4) =2/5ζ(2)^2 and ζ(6) =8/35ζ(2)^3. More generally, it shows that ζ(2) is a rational multiple of ζ(2) n. These results were rediscovered and extended by Ramanujan [11] and many others [1][5][6][8].


Item Type:Book Section
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https://doi.org/10.1007/978-1-4615-0304-0_14DOIArticle
Additional Information:© Springer Science+Business Media New York 2003. To Solomon Golomb for his seventieth birthday.
Series Name:Springer International Series in Engineering and Computer Science
Issue or Number:726
Record Number:CaltechAUTHORS:20190909-133031057
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190909-133031057
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98521
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:09 Sep 2019 21:33
Last Modified:03 Oct 2019 21:41

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